Guy Metcalfe

Experimental and computational studies of mixing in complex Stokes flows: the vortex mixing flow and multicellular cavity flows

A complex Stokes flow has several cells, is subject to bifurcation, and its velocity field is, with rare exceptions, only available from numcrical computations. We present experimental and computational studies of two new complex Stokes flows: a vortex mixing flow and multicell flows in slender cavities. We develop topological relations between the geometry of the flow domain and the family of physically realizable flows; we study bifurcations and symmetries, in particular to reveal how the forcing protocol’s phase hides or reveals symmetries. Using a variety of dynamical tools, comparisons of boundary integral equation numerical computations to dye advection experiments are made throughout. Several findings challenge commonly accepted wisdom. For example, we show that higher-order periodic points can be more important than period-one points in establishing the advection template and extended regions of large stretching. We demonstrate also that a broad class of forcing functions produces the same qualitative mixing patterns. We experimentally verify the existence of potential mixing zones for adiabatic forcing and investigate the crossover from adiabatic to non-adiabatic behaviour. Finally, we use the entire array of tools to address an optimization problem for a complex flow. We conclude that none of the dynamical tools alone can successfully fulfil the role of a merit function; however, the collection of tools can be applied successively as a dynamical sieve to uncover a global optimum.

Isolated Mixing Regions : Origin, Robustness and Control

Isolated mixing regions (IMRs) are fluid regions that may or may not have interior mixing and are usually located far away from boundaries; however, they do not exchange material with regions of active global mixing and they therefore present a substantial obstacle to global mixing. Islands in two dimensions and tubes and tori in three dimensions are examples of IMRs. We investigate interrelated issues dealing with IMR detection and prediction, robustness, exploitation and control of IMRs using experiments and computations carried out in two model systems. Results indicate that it is possible to manipulate the area of IMRs in a controlled fashion, hence creating an analog of a controlled release capsule within a chaotic flow.

Kinematic considerations for mantle mixing

Recent experimental and computational studies tal and computational mixing studies in highly controlled show thatislands(u~mixedrcgions that donot interactwith flows show that even seemingly inconsequential errors in the surrounding regions) are ubiquitous features in chaotically computed velocity field of 3-D or time-dependent flows can advecting fluids. Such is~ands~~uitenaturall~ account for the geochemically inferred coexistence of apparently distinct, long-lived geochemical heterogeneity with relatively homo- geneous regions of an actively convecting mantle. These results also indicate that mixing patterns-the set of islands and folds characterizing the large-scale material advection- are sensitive to small variations in the rheology of the fluid. Therefore, interpretation of numerical simulations of mantle transport and mixing is less straightforward than currently supposed. Computational studies of analytic flow solutions with systematically introduced and controlled errors indi- cate that mantle simulations are unlikely to accurately com- pute individual trajectories for even moderate time, but that tmjectory ensembles can be accuntely computed for long time. Significantly, computations also indicate that mixing and transport resultsmay not evolve smoothly with increased rheological realism.

Chaotic mixing processes: New problems and computational issues

Abstract As mixing problems evolve beyond purely kinematic concerns, new issues appear. One issue is that analyses must often be based on numerical solutions of the Navier-Stokes, or more complex, equations; a second is the ability to deal with complexities involving the coupling of local and global dynamics, as occur, for example, in problems of aggregation and breakup. Both aspects are briefly considered, the bulk of the comments pertaining primarily to intrinsic limits of mixing simulations.

Convection in 3 He–superfluid- 4 He mixtures. Part 2. A survey of instabilities

Dilute mixtures of 3He in superfluid 4He have Prandtl numbers easily tunable between those of liquid metals and water: 0.04 < Pr < 2. Moreover, superfluid mixture convection is closely analogous to classical Rayleigh-Bhard convection, i.e. superfluid mixtures convect as if they were classical, single-component fluids. This work has two goals. The first, accomplished in Part 1, is to experimentally validate the superfluid mixture convection analogue to Rayleigh-BCnard convection. With superfluid effects understood and under control, the second goal is to identify and characterize time-dependence and chaos and to discover new dynamical behaviour in strongly nonlinear convective flows. In this paper, Part 2, we exploit the unique Pr range of superfluid mixtures and the variable aspect ratio (r) capabilities of our experiment to survey convective instabilities in the broad, and heretofore largely unexplored, parameter space 0.12 < Pr < 1.4 and 2 < r < 95. Within this large parameter space, we have focused on small to moderate r and Pr and on large r with Pr w 1. The novel behaviour uncovered in the survey includes the following. Changing attractors: at r = 6.0 and Pr = 0.3, we observe intermittent bursting destabilizing a fully developed chaotic state. Above the onset of bursting the average length of a burst-free interval and the average length of a burst vary as power laws. At r = 4.25 and Pr = 0.12 we observe a particularly novel reversible switching transition involving two chaotic attractors. Instability competition: near the codimension-2 point at the crossing of the skewed-varicose and oscillatory instabilities we find that the effects of instability competition greatly increase the complexity and multiplicity of states. A heat-pulse method allows selection of the active state. Decreasing r suppresses the available complexity. Superfluid turbulence : we find that the large-amplitude noisy states, previously believed due to superfluid turbulence, are confined to small values of r and Pr and are not consistent with superfluid turbulence. Changing instabilities: at Pr = 0.19 a wavevector detuning changes the type of secondary instability from oscillatory to saddle-node, with an unusual 3/4 exponent time scaling. Very large r: at Pr = 1.3 for r increasing from 44 to 90, we observe the onset of convection changing from ordered and stationary to disordered and time-dependent. At the beginning of the crossover there are hysteretic transitions to coherent oscillations close to the onset of convection. By the end of the crossover convection is timedependent and irregular at onset with the fluctuation amplitude correlated with the mean Nusselt number.

Convection in 3He-superfluid-4He mixtures. Part 1. A Boussinesq analogue

Dilute mixtures of 3 He in superfluid 4 He have Prandtl numbers easily tunable between those of liquid metals and water : 0.04 < Pr < 2. Moreover, owing to the tight coupling of the temperature and concentration fields, superfluid mixture convection is closely analogous to classical Rayleigh-Benard convection, i.e. superfluid mixtures convect as if they were classical, single-component fluids, well described by the Boussinesq equations. This work has two goals. The first is to put the theory of superfluid mixture convection on a firmer basis. We accomplish this by combining experiment and analysis to measure superfluid effects on the onset of convection. In the process, we demonstrate quantitative control over superfluid effects and, in particular, that deviations from classical convective behaviour can be made small or at worst no larger than finite aspect ratio effects. The size of superfluid effects at convective onset can be less than a few percent for temperatures 1 < T < 2 K. Comparison of the measured properties of superfluid mixture roll instabilities above the onset of convection to the properties predicted by Boussinesq calculations further verifies that superfluid mixtures convect as classical fluids.